Friday, April 28, 2017

The science-fiction writer Robert
Anton Wilson invented a word, sumbunol,
meaning “some but not all”. He did so to combat the temptation to
over-generalize. Sumbunol’s formal definitions are:

Sumbunol(x)(P(x))

=Exists(x)(P(x))andExists(y)(not P(y))

=Exists(x,y)( P(x) xor P(y) )

The last equation says; “sumbunol
things are P” is equivalent to “P differs on some two things”. Sumbunol = varying; so another symbol or sumbunol
is “Var”: Var(x)(P(x))=P varies.

The negation of sumbunol is ollerno, meaning “all or no”, with these
formal definitions:

Ollerno(x)(P(x))

=All(x)(P(x))orAll(y)(not P(y))

=All(x,y)( P(x) iff P(y) )

The last equation says; “ollerno
things are P” is equivalent to “P is the same for any two things”.Ollerno = constant;
so another symbol for ollerno is “Con”: Con(x)(P(x)) = P is constant.

Given
a universe of discourse with two elements {a,b}, and a predicate P(x) on {a,b},
then:

All(x)(P(x))=P(a) and P(b)

Some(x)(P(x))=P(a) or P(b)

No(x)(P(x))=not (P(a) or P(b))

NotAll(x)(P(x))=not(P(a) and P(b))

Con(x)(P(x))=P(a) iff P(b)

Var(x)(P(x))=P(a) xor P(b)

So these six quantifiers correspond to the six non-constant
commutative boolean functions on two inputs.

If
the universe of discourse has only one element {a}, then:

All(P)=Some(P) = P(a)

NotAll(P) = No(P)=not P(a)

Var(P)=F

Con(P)=T

And if the universe of discourse is empty, then:

All(P)=T

Some(P) =F

NotAll(P) = F

No(P)=T

Var(P)=F

Con(P)=T

Over
a larger universe of discourse {a1, a2, a3…}, define:

All(x)(P(x))=P(a1) and P(a2) and P(a3) and …

Some(x)(P(x))=P(a1) or P(a2) or P(a3) or …

No(x)(P(x))=not ( P(a1) or P(a2) or P(a3)
or … )

NotAll(x)(P(x))=not
( P(a1) and P(a2) and P(a3) and … )

Con(x)(P(x))= All(x)(P(x)) or No(x)(P(x))

Var(x)(P(x))= Some(x)(P(x)) and NotAll(x)(P(x))

In general these equations hold:

Negation:

Var(x)(P(x))=Var(x)(not P(x))=not Con(x)(P(x))

Con(x)(P(x))=Con(x)(not P(x))=not Var(x)(P(x))

Partial
Distribution:

A
and Var(x)(P(x))=Var(x)( A and P(x) )

A or Con(x)(P(x))=Con(x)( A or P(x) )

“And” distributes over sumbunol, and
“or” distributes over ollerno; but “and” does not distribute over ollerno; nor
does “or” distribute over sumbunol:

TrueorVar(P(x)) = True;but Var( True or P(x) ) = False.

False and Con(P(x)) = False;butCon( False and P(x) ) = True.

Sumbunol and ollerno have these Equivalence Rules:

Con(x)( P(x) iff Q(x) )andCon(x)(Q(x) iff R(x))

ImpliesCon(x)(P(x) iff R(x))

If
“P iff Q” and “Q iff R” are constant, then “P iff R” is constant.

Con(x)( P(x) iff Q(x) )andCon(x)(Q(x))

ImpliesCon(x)(P(x))

If
“P iff Q” is constant, and Q is constant, then P is constant.

Var(x)(P(x))andCon(x)(Q(x))

impliesVar(x)( P(x) iff Q(x) )

If P varies and Q is constant, then “P
iff Q” varies.

Con(x)( P(x) iff Q(x) )

ImpliesCon(x)(P(x))iffCon(x)(Q(x))

If “P iff Q” is constant, then P and Q
are equally constant.

Var(x)(P(x))xorVar(x)(Q(x))

impliesVar(x)(
P(x) xor Q(x) )

If
P varies or else Q varies, then “P or else Q” varies.

Sumbunol and ollerno also have two Functionality Rules:

If
F(p,q) is any function on Boolean logic, then

Con(x)(P(x))andCon(x)(Q(x))

ImpliesCon(x) ( F(P(x),Q(x)) )

This is “Constancy”: constant inputs imply a constant output.

If
F(p,q) is any function on Boolean logic, then

Var(x) ( F(P(x),Q(x)) )

ImpliesVar(x)(P(x))orVar(x)(Q(x))

This is “Variability”: varying output implies a varying input.

Here is “Proof By Constancy Plus Example”:

For all a and b,

Con(x)(P(x))andP(a)

ImpliesP(b)

Here are “Variation By Opposing Examples”:

For all a and b,

P(a)andnot P(b)

implies Var(x)(P(x))

Here is “Constancy and Existence implies Universality”:

Con(x)(P(x))andExists(x)(P(x))

impliesAll(x)(P(x))

Here is “Existence implies Variation or Universality”:

Exists(x)(P(x))

implies Var(x)(P(x))orAll(x)(P(x))

The reverse implications require that
the universe of discourse of the quantifiers be not empty; i.e. that something exists:Exist(x)(x=x)

IfExist(x)(x=x),then

All(x)(P(x))iffCon(x)(P(x))andExists(x)(P(x))

Exists(x)(P(x))iffVar(x)(P(x))orAll(x)(P(x))

No(x)(P(x))iffNotAll(x)(P(x))andCon(x)(P(x))

NotAll(x)(P(x))iffVar(x)(P(x))orNo(x)(P(x))

Var(x)(P(x))iffSome(x)(P(x))andNotAll(x)(P(x))

Con(x)(P(x))iffAll(x)(P(x))orNo(x)(P(x))

If
anything exists, then

universality = constancy and
existence

existence = variation or universality

nonexistence
= exceptions and constancy

exceptions
= variability or nonexistence

variability
= existence and exceptions

constancy=universality or nonexistence

Here is a Wilsonian Quantifier Troika:

Moe: No frogs are princes.

Larry: Some but not all frogs are
princes.

Curly: All frogs are princes.

Moe,
Larry and Curly all agree that frogs exist.

By
2/3 majorities each, we get this Wilsonian
Trilemma:

LK: Some frogs are princes.

ML: Some frogs are not princes.

KM: All or no frogs are princes.

The
last can be read, “All frogs are equally
princes.”

Any two of a trilemma imply the
negation of the third. Therefore:

If some frogs are princes, and some
frogs are not princes, then sumbunal frogs are princes.

If some frogs are not princes, and
ollerno frogs are princes, then no frogs are princes.

If ollerno frogs are princes, and
some frogs are princes, then all frogs are princes.

In
general a Wilsonian trilemma has the form:

Some A have property P;

Some A do not have property P;

All As have property P equally.

For
instance:

Some men are good;

Some men are not good;

All men are equally good.

The
trilemma implies these three deduction rules:

If some men are good, and some men
are not good,

then
not all men are equally good.

If some men are
not good, and all men are equally good,

then no men are good.

If all men are equally good, and some
men are good,

then
all men are good.

Here are Wilsonian versions of mathematical induction:

All(n)( P(n) iff P(n+1) )=Con(n)( P(n) )

Var(n)(
P(n) )=Some(n)( P(n) xor P(n+1) )

On the integers, the iffs and xors of ollerno and sumbunol
need only be between elements separated by adding one. The integers are
deductively linked by succession.

In nonstandard analysis, where there are infinitesimal
quantities, you can express the intermediate value theorem in Wilsonian terms:

If f(x) is continuous on [a,b], and i is any infinitesimal,
then

Con(x)(
f(x)>0 )=All(x) ( f(x)>0ifff(x+i)>0 )

f’s sign is constant
if it is constant under any infinitesimal change.

Var(x)(
f(x)>0 )=Some(x) ( f(x)>0xorf(x+i)>0 )

f’s sign varies if it
varies under some infinitesimal change.

Wilsonian
quantifiers even apply to hashtag politics. Consider the hashtag #BLM = “Black
Lives Matter”. This hashtag denotes an aspiration, not a description. Its
implicit protest message is that as things are, black lives do not matter.

A Wilsonian hashtag would be: #SBNALM = “Some but not all
lives matter”. That is a cynical description of political reality. Its
aspirational opposite: #ALMOND = “All lives matter or none do”.

Actually, I think that, in the long run, #SBNALM is an
overclass aspirational delusion, and #ALMOND is the gritty reality.

Thursday, April 27, 2017

This
paradox unites the paradox of the boundary with the paradox of infinite parity.

It’s
a sunny day at the frog pond, but a large tree is casting a dark shadow. A frog
is sitting a foot into the shade. Feeling too cold, the frog jumps out of the
shade and lands 1/2 foot into the bright sunlight. Feeling too warm, the frog
jumps out of the sunlight and lands 1/4 foot into the shade. Feeling too cold,
the frog jumps out of the shade and lands 1/8 foot into the bright sunlight.
Feeling too warm, the frog jumps out of the sunlight and lands 1/16 foot into
the shade. And so on; moreover, the frog’s jumps accelerate geometrically, so
they are all done in finite time.

When
the frog finally settles on the shade line, it has alternated between warm and
cold infinitely many times. On the shade line, is the frog warm or cold?

If,
like Baby Bear’s porridge, the frog is ‘just right’, then that’s a third value
in addition to ‘warm’ or ‘cold’. The third value arises from an infinite
alternation of the other two; so this is like asking if infinity is odd or
even.

Now
let us take into account the movement of the shadow. Upon reaching the shade
line, the frog enters Samadhi; but then awakes an hour later to find the shade
line shifted. It then makes another infinity of jumps, reaches the shade line
again, and goes back to sleep. Then the shadow-line moves, and the process
repeats.

By
appropriately adjusting the frog’s times of waking and sleeping, we can make
the frog go through any countable infinite ordinal sequence of jumps in the
course of the day. Let the number of frog jumps by the end of the day to be a
large countable infinite ordinal; large enough, say, to exceed any recursive
ordinal naming scheme. All though that busy day the frog was warm, cold or
asleep. At the moment of sunset, is the frog awake or asleep? And if awake,
warm or cold?